The negative answer to decidable = non-contracting grammar? suggests the following question:

Is there a decidable language that can be recognized only by a space unrestricted Turing Machine (i.e. with infinite tape but in finite time)?

This is, are there words, w, in a decidable language for which cannot be determined a bound f(|w|)?

  • 1
    $\begingroup$ What is a restricted turing machine? $\endgroup$
    – nir shahar
    May 2 at 10:37
  • $\begingroup$ @nirshahar It is a Turing machine with an upper bound on the amount of tape it can use, related to length of its input. For instance, languages by type-1 grammars, can be recognized by a linear bounded Turing Machine (they don't need more thant k |w| space to accept word w). $\endgroup$ May 2 at 10:49
  • $\begingroup$ As Yuval has pointed, for a decidable language L, f() can always computed. If $M_L$ is the deciding TM for L, then f() can be calculated for a given length n just by taking the longest time/space used for $M_L(w)$ where |w| = n. $\endgroup$ May 3 at 9:46

Consider a Turing machine which reads its input and then immediately stops. This Turing machine always halts on every input, but the running time is unbounded: the machine runs for $n$ steps on an input of length $n$.

In fact, most halting Turing machines have unbounded running time: if the running time of a Turing machine is bounded by $T$, then the language it decides can only depend on the first $T$ symbols of the input.

If a Turing machine always halts on any input, then there is a function $f(n)$ such that the Turing machine always halts within $f(n)$ steps on an input of length $n$. You can take as $f(n)$ the maximal time it takes the Turing machine to halt on an input of length $n$. Since there are only finitely many words of length $n$ (recall that the input alphabet is fixed), this is well-defined.

Furthermore, if a Turing machine always halts on any input, the function $f(n)$ defined in the preceding paragraph is always computable. You can compute it by simulating the machine on all possible inputs of length $n$.

Space works exactly the same way.

  • $\begingroup$ Mmmm, sorry now I'm confused. Just to clarify: are your saying that for any decidable language there is a stoping Turing Machine with space bounded by some suitable function f(n) where n is the length of the input? $\endgroup$ May 2 at 17:24
  • $\begingroup$ For your sentence "This Turing machine always halts on every input, but the running time is unbounded: the machine runs for n steps on an input of length n" I would say that the running time is bounded O(n), thus not really unbounded. Thus, what I'm trying to ask if every stoping Turing Machine has a O(f(n)) space (or time). Maybe I'm asking a nonsense. Sorry. $\endgroup$ May 2 at 17:33
  • $\begingroup$ If a language is decidable, then by definition there is a Turing machine that decides it. This Turing machine runs in time $f(n)$, where $f(n)$ is defined in my answer. Similarly, it uses space $g(n)$, where $g(n)$ is the maximum space it uses on all (finitely many) words of length $n$. $\endgroup$ May 2 at 17:37
  • $\begingroup$ So the answer to the question title is YES. Right? $\endgroup$ May 3 at 9:18
  • $\begingroup$ Your "restricted Turing machine" is just a Turing machine which halts on every input. $\endgroup$ May 3 at 9:22

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